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We show that there are strictly pseudoconvex, real algebraic hypersurfaces in$\bC^{n+1}$ that cannot be locally embedded into a sphere in $\bC^{N+1}$ forany $N$. In fact, we show that there are strictly pseudoconvex, real algebraichypersurfaces in $\bC^{n+1}$ that cannot be locally embedded into any compact,strictly pseudoconvex, real algebraic hypersurface.

On the existence of holomorphic embeddings of strictly pseudoconvex algebraic hypersurfaces into spheres